resonances from amplitudes in sliced bins in energy
The two reactions and are analyzed to determine the leading photoproduction multipoles and the pion-induced partial wave amplitudes in slices of the invariant mass. The multipoles and the partial-wave amplitudes are simultaneously fitted in a multichannel Laurent+Pietarinen model (L+P model), which determines the poles in the complex energy plane on the second Riemann sheet close to the physical axes. The results from the L+P fit are compared with the results of an energy-dependent fit based on the Bonn-Gatchina (BnGa) approach. The study confirms the existence of several poles due to nucleon resonances in the region at about 1.9 GeV with quantum numbers , .
The nucleon and its excited states are the simplest systems in which the non-abelian character of strong interactions is manifest. Three quarks is the minimum quark content of any baryon, and these three quarks carry the three fundamental colour charges of Quantum Chromodynamics (QCD), and combine to a colourless baryon. At present it is, however, impossible to calculate the spectrum of excited states from first principles, even though considerable progress in lattice gauge calculations has been achieved Edwards:2011jj . Models are therefore necessary when data are to be compared to predictions.
Quark models predict a rich excitation spectrum of the nucleon Capstick:1986bm ; Ferraris:1995ui ; Glozman:1997ag ; Loring:2001kx ; Giannini:2015zia . In quark models, the resonances are classified in shells according to the energy levels of the harmonic oscillator. The shell structure of the excitations is still seen in the data and reproduced in lattice calculations Edwards:2011jj . The first excitation shell is predicted to house five and two resonances with negative-parity; all of them are firmly established. The second excitation shell contains missing resonances: 22 resonances (14 ’s and 8 ’s) are predicted but 15 only are found in the mass range below 2100 MeV, and just 10 of them (5 ’s and 5 ’s) are considered as established, with three or four stars in the notation of the Particle Data Group Olive:2016xmw . Thus 9 ’s in the mass region between 1700 MeV and 2100 MeV are predicted to exist which are unobserved or the evidence for their existence is only fair or even poor. This deficit is known as the problem of the missing resonances Koniuk:1979vw ; Koniuk:1979vy . The search for missing resonances is one of the major aims of a number of experiments in which the interaction of a photon beam in the GeV energy range with a hydrogen and deuterium target is studied.
In elastic and charge exchange scattering, the excited states may have isospin () and (). A large amount of data was analyzed by the groups at Karlsruhe-Helsinki (KH84) Hohler:1984ux , Carnegie-Mellon (CM) Cutkosky:1980rh and at GWU Arndt:2006bf . The 1850 - 2100 MeV mass region – where the missing resonances of the second excitation shell are predicted in most constituent quark models (see, e.g., Capstick:1986bm ; Ferraris:1995ui ; Glozman:1997ag ; Loring:2001kx ; Giannini:2015zia ; Capstick:1998uh ; Capstick:2000qj ) – is dominated by the production of resonances with spin-parity , ; nucleon resonances are difficult to establish in this mass range due to the overwhelming background from resonances.
The production of hyperons in pion and photo-induced reactions, in contrast to elastic scattering, is ideally suited to search for new nucleon resonances and to confirm resonances that are not yet well established (see, e.g., Capstick:1998uh ; Capstick:2000qj and references therein). Due to isospin conservation in strong interactions, only resonances decay into final states, there are no isospin contributions. Second, the weak-interaction decay reveals the polarization of the . Thus, the recoil polarization is measurable. In elastic scattering, the equivalent target polarization, also called , requires the use of a polarized target. In photoproduction, a third advantage emerges: the process is not suppressed even when the coupling constants of resonances in the second excitation shell are small Capstick:1998uh ; Capstick:2000qj . Photoproduction may hence reveal the existence of resonances coupling to only weakly. Indeed, a number of new resonances has been reported (or have been upgraded in the star rating) from a combined analysis of a large number of pion and photo-produced reactions Anisovich:2011fc . Some of the “new” resonances had been observed before Hohler:1984ux ; Cutkosky:1980rh ; Manley:1992yb ; Penner:2002ma ; Penner:2002md or were confirmed in later analyses Shrestha:2012va ; Shrestha:2012ep ; L+P2014 . The evidence for the existence of the new states stemmed from energy-dependent fits to the data using the BnGa approach Anisovich:2004zz ; Anisovich:2006bc ; Anisovich:2007zz ; Denisenko:2016ugz . The reaction proved to be particularly useful Nikonov:2007br .
The ultimate aim of experiments is to provide sufficient information that the data can be decomposed into partial waves or multipoles of defined and unique spin-parity. It can either be done through constructing an explicit theoretical model, or as we present here, through the reconstruction of partial-wave amplitudes and of multipoles in a truncated partial wave analysis. Limiting the partial wave series to low orbital angular momenta allows us to overcome issues with the still relatively large errors in the measurements of observable quantities.
The main goal of this paper is to test if resonances in the fourth resonance region can be confirmed definitely from a fit to multipoles driving the excitation of partial waves with defined spin-parity, and to extract their properties. This is done in two ways: i.) In a standard way where a theoretical model is constructed. Its free parameters are estimated by fitting to the experimental data set base, and the partial waves of the final solution are analytically continued into the complex energy plane to obtain poles. ii.) In a way which does not depend on detailed model assumptions by using the Laurent+Pietarinen (L+P) method where a solution of the theoretical model is replaced by a most general analytic function consisting of a number poles and branch-cuts, which is embodied by a fast converging power series in a conformal variable. This variable is generated by a conformal mapping of the complex energy plane onto a unit circle. The first Riemann sheet is mapped to the outside of the unit circle, and the second Riemann sheet – where the poles are located – into the inside of the unit circle. In method ii.), poles are extracted by fitting to the single-energy partial wave decomposition, as opposed to a direct global fit to the data.
Method i.), coupled-channel energy-dependent fits, exploits the full statistical potential of the data. The effect of couplings to various other final states like , , , , etc. is taken into account exactly as well as all correlations between the different amplitudes. However, all partial waves need to be determined in one single fit, and it is difficult to verify the uniqueness of the results. In method ii.) we use single channel L+P fits (SC L+P) where each channel is fitted individually, and multi-channel L+P fits (MC L+P) where two or more channels are fitted simultaneously. The main advantage of the model-independent approach is that we can fit one partial wave at a time, and that we avoid any dependence on the quality of the model. The drawback is that you first have to extract partial waves, and this procedure depends on the choice of higher partial waves, introducing some model dependence.
2 Construction of amplitudes in slices of their invariant mass
2.1 The partial wave amplitudes for
The differential cross sections for the reaction
receives contributions from a spin-non-flip and a spin-flip amplitude, and , according to the relation
where and are the initial and final meson momenta respectively in the centre of mass frame Hohler:1984ux . Both amplitudes depend on the invariant mass and , with being the scattering angle. The two amplitudes can be expanded into partial wave amplitudes
where are the Legendre polynomials. is the total spin of the state. The sign in the relation for defines the sign in .
The decay can be used to determine the decay asymmetry with respect to the scattering plane, called recoil asymmetry . Assuming that the target nucleon is fully polarized, can be defined as
When the target proton is polarized longitudinally (along the pion beam line), the spin transfer from proton to yields the spin rotation angle .
It is defined as , where and are the polarization components in direction of the and its orthogonal component in the scattering plane. and are given by
The polarization variables are constrained by the relation
2.1.2 Fits to the data
Data on the reaction were taken in Chicago Knasel:1975rr and at the NIMROD accelerator at the Rutherford Laboratory Baker:1978qm ; Saxon:1979xu ; Bell:1983dm . From these data, the partial wave amplitudes defined in eqn. (2) should be derived.
A detailed study showed that the data require angular momenta up to or even but do not have the precision to determine all partial wave amplitudes Anisovich:2014yza . Therefore we try to determine at least the low- amplitudes, in particular , , , leading to , , and . The higher partial waves, those above , , , are taken from our current BnGa fit.
Figure 1 shows the data. The solid curves represent the final BnGa fit. It reproduces the data with a . The number of free parameters is 75.
The fit returns the real and imaginary parts of amplitudes for the , , and partial waves. The and amplitudes are shown in Fig. 2, the amplitude in Ref. Anisovich:2014yza only (since it could not be fit with the L+P method). The solid line represents the L+P fit described below, and the energy-dependent solution BnGa2011-02 is shown as error band. Note that the higher partial waves are constrained fixed to the BnGa solution, while the other lower amplitudes are free to adopt any values.
2.2 The multipoles for
The amplitude for the reaction can be written in the form
where and are spinors representing the baryon in the initial and final state, is the electromagnetic current of the nucleon, and characterizes the polarization of the photon. The amplitude can be expanded into four invariant (CGLN) amplitudes Chew:1957tf
where is the momentum of the hyperon in the final state, is the momentum of the nucleon in the initial state, calculated in the center-of-mass system of the reaction, and are the Pauli matrices. These four functions are functions of the invariant mass and of with and as the scattering angle. A determination of these four amplitudes requires the measurement with sufficient accuracy of at least eight well chosen observables Barker:1975bp ; Fasano:1992es ; Keaton:1996pe ; Chiang:1996em ; Sandorfi:2010uv . For each slice in energy and angle one phase remains undetermined. It needs to be fixed from other sources. In elastic scattering, the phase can be determined from the (calculable) Coulomb interference. In hyperon production, one could try to fix the phase to the phase of -channel Kaon exchange. Once the functions are known for each energy and angle, the results of all experiments can be predicted.
The relations between the functions and the observables can be found, e.g.,
in Sandorfi:2010uv . For convenience, we give the expressions for the observables used
in the fits. The differential cross section and the
single polarization observables, the beam asymmetry , the recoil asymmetry , and the target asymmetry , are given by
The double polarization observables , (, ) define the spin transfer from linearly (circularly) polarized photons to the hyperon where the axis is given by the meson direction. This is referred to as the primed frame. Experimentally, the data on the spin transfer from polarized photons to the hyperon are sometimes presented in an unprimed frame, in which the photon momentum is chosen as reference axis. Observables in the two frames are related by a simple rotation:
with similar relations holding for the quantities and .
The double polarization observables , (, ) can be written as
When the are known with sufficient statistical accuracy they can be expanded – for each slice in energy – into power series using Legendre polynomials and their derivatives:
Here, corresponds to the orbital angular momentum in the system, is the total energy, are Legendre polynomials with , and and are electric and magnetic multipoles describing transitions to states with . or multipoles do not exist. Processes due to meson exchanges in the channel may provide significant contributions to the reaction. They may demand high-order multipoles. The minmal required to describe the data can be determined by polynomial expansions of the data Wunderlich:2016imj . A more direct approach is to insert the functions (eqns. 10) into the expressions for the observables (eqns. 9a and 9f) and to truncate the expansion at an appropriate value of Wunderlich:2014xya . The observables are now functions of the invariant mass and the scattering angle, and the fit parameters are the electric and magnetic multipoles. In this method, the number of observables required to get the full information might be reduced if the number of contributing higher partial waves is not too big. But still, high precision is mandatory for the expansion.
2.2.2 Fits to the data
From the results of the BnGa analysis we expect that in the energy range considered here the , , and yield the largest contributions, followed by , , and . The , , , , , , , all contribute with increasingly smaller importance, higher multipoles become negligible. First fits showed that it is not possible, given the statistical and systematic accuracy of the data, to determine all significant partial waves. Due to strong correlations between the parameters, the errors became large and the resulting multipoles showed large point-to-point fluctuations. Hence we decreased the number of freely fitted multipoles; the higher multipoles were fixed to the BnGa results. These multipoles are shown in Fig. 3. Reasonably small errors were obtained when the four multipoles , , , and were fitted. The errors increased only slightly when the multipoles , , and were fitted in addition but constrained to the BnGa solution by a penalty function.
where and are the electric and magnetic multipoles from solution with , , and fitted freely; , are the multipole errors.
The reaction has been studied extensively by the CLAS collaboration. The early measurement of the differential cross sections Bradford:2005pt was later superseded by a new measurement reporting the differential cross sections and the recoil polarization McCracken:2009ra . The spin transfer from circularly polarized photons to the final-state hyperon, the quantities and , were reported in Bradford:2006ba . The polarization observables have been determined recently Paterson:2016vmc . The data are shown in Figs. 4-6. The data are used to determine the photoproduction multipoles in a truncated partial wave analysis.
The final result for the multipoles are shown in Fig. 9. Strong variations are observed. The imaginary parts of all multipoles, except and , show threshold enhancements due to (), (), ( and ), (). Further structures are clearly seen at about 1900 MeV in the , , , , , multipoles.
These structures emerge reliably when the multipole series is truncated, and only few multipoles are fitted freely. In Fig. 9 we show the results from one of our tests. In this case, the seven largest multipoles, , , , , , , and were all left free. In several mass bins, the resulting multipoles show an erratic behavior; the results become unstable. Likewise, it was important to include the multipoles with large orbital angular momenta. Even though they are individually all small, neglecting them (by assuming that they are identically zero) leads to biased results. Furthermore, these multipoles fix the overall phase.
Sandorfi, Hoblit, Kamano, and Lee Sandorfi:2010uv have reconstructed the photoproduction amplitudes for the reaction . For the high partial waves, they used the Born amplitude. Partly, they fitted all waves with freely and determined the phases as differences to the phase. In other fits, they had the phase free and fitted all waves with . The resulting multipoles showed a wide spread. They concluded that a very significant increase in solid-angle coverage and statistics is required when all partial waves up to are to be determined.
3 BnGa fits to the data
The BnGa partial wave analysis uses a matrix formalism to fit data on pion and photo-induced reactions to extract the leading singularities of the scattering or production processes. The formalism is described in detail in a series of publications Anisovich:2004zz ; Anisovich:2006bc ; Anisovich:2007zz ; Denisenko:2016ugz . Here we briefly outline the dynamical part of the method.
The pion induced reaction from the initial state to the final state is described by a partial wave amplitude . It is given by a -matrix which incorporates a summation of resonant and non-resonant terms in the form
The multi-index denotes the quantum numbers of the partial wave, it is suppressed in the following. The factor represents the phase space matrix to all allowed intermediate states, , are the phase space factors for the initial and the final state. The matrix parametrizes resonances and background contributions:
Here are coupling constants of the pole to the initial and the final state. The background terms describe non-resonant transitions from the initial to the final state.
For photoproduction reactions, we use the helicity ()-dependent amplitude for photoproduction of the final state Chung:1995dx
is the photo-coupling of a pole and a non-resonant transition. The helicity amplitudes , are defined as residues of the helicity-dependent amplitude at the pole position and are complex numbers Workman:2013rca .
In most partial waves, a constant background term is sufficient to achieve a good fit. Only the background in the meson-baryon -wave required a more complicated form:
Further background contributions are obtained from the reggeized exchange of vector mesons Anisovich:2004zz in the form
here, represents a vertex function and a form factor. describes the trajectory, , is a normalization factor, and the signature of the trajectory. Pion and and Pomeron exchange both have a positive signature and therefore Anisovich:2004zz :
Additional -functions eliminate the poles at :
where the Kaon trajectory is parametrized as , with given in GeV.
The data on partial wave amplitudes (Fig. 2) and on the photoproduction multipoles (Fig. 9) were included in the data base of the BnGa partial wave analysis. The data are fitted jointly with data on , , , , and from both photo- and pion-induced reactions. Thus inelasticities in the meson-baryon system are constrained by real data. A list of the data used for the fit can be found in Anisovich:2011fc ; Anisovich:2013vpa ; Sokhoyan:2015fra ; Gutz:2014wit and on our website (pwa.hiskp.uni-bonn.de). In Fig. 2, the systematic errors define the error band; in Fig. 9, the systematic error of the real (imaginary) part of the amplitudes is shown a grey (red/blue) histogram at the bottom (top) line. The systematic errors are derived by a variation of the model space by adding further resonances with different spin-parities when the data are fitted.
4 The Laurent-Pietarinen expansion
The main task of the single channel Laurent-Pietarinen expansion () is extracting pole positions from given partial waves for one reaction. The driving concept behind the method is to replace an elaborate theoretical model by a local power-series representation of partial wave amplitudes L+P2013 . The complexity of a partial-wave analysis model is thus replaced by much simpler model-independent expansion which just exploits analyticity and unitarity. The L+P approach separates pole and regular part in the form of a Mittag-Leffler expansion111Mittag-Leffler expansion Mittag-Leffler is the generalization of a Laurent expansion to a more-than-one pole situation. From now on, for simplicity, we will simply refer to this as a Laurent expansion., and instead of modeling the regular part using some physical model it uses the conformal-mapping-generated, rapidly converging power series with well defined analytic properties called a Pietarinen expansion222A conformal mapping expansion of this particular type was introduced by Ciulli and Fisher Ciulli ; CiulliFisher , was described in detail and used in pion-nucleon scattering by Esco Pietarinen Pietarinen ; Pietarinen1 , and named as a Pietarinen expansion by G. Höhler in Hohler:1984ux . to represent it effectively. So, the method replaces the regular part calculated in a model with the simplest analytic function which has correct analytic properties of the analyzed partial wave (multipole), and fits the given input. In such an approach the model dependence is minimized, and is reduced to the choice of the number and location of L+P branch-points used in the model. The method is applicable to both, theoretical and experimental input, and represents the first reliable procedure to extract pole positions directly from experimental data, with minimal model bias. The L+P expansion based on the Pietarinen expansion is used in some former papers in the analysis of pion-nucleon scattering data Ciulli ; CiulliFisher ; Pietarinen ; Pietarinen1 and in several few-body reactions L+P2014 ; L+P2014a ; L+P2015 . The procedure has recently been generalized also to the multi-channel case Svarc2016 .
The generalization of the L+P method to a multichannel L+P method, used in this paper, is performed in the following way: i) separate Laurent expansions are made for each channel; ii) pole positions are fixed for all channels, iii) residua and Pietarinen coefficients are varied freely; iv) branch-points are chosen as for the single-channel model; v) the single-channel discrepancy function (see Eq. (5) in ref. L+P2015 ) which quantifies the deviation of the fitted function from the input is generalized to a multi-channel quantity by summing up all single-channel contributions, and vi) the minimization is performed for all channels in order to obtain the final solution.
The formulae used in the L+P approach are collected in Table 1.
L+P is a formalism which can be used for extracting poles from any given set of data, either theoretically generated, or produced directly from experiment. If the data set is theoretically generated, we can never reconstruct the analytical properties of the background put into the model, we can only give the simplest analytic function which on the real axes gives very similar, in practice indistinguishable result from the given model values. Therefore, analyzing partial waves coming directly from experiment is for L+P much more favourable because we do not have such demands. The analytic properties are unknown, so there is no reason why the simplest perfect fit we offer should not be the true result. As in principle we do not care whether the input is generated by theory or otherwise, in the set of formulas given in Table 1. we denote any input fitted with L+P function generically as .
In this paper we fit partial wave data; the discrete data points coming from a semi-constrained single energy fit of K photo-production data, which is obtained in a way that the partial waves with are fixed to Bonn-Gatchina energy dependent partial waves, and lower ones are allowed to be free. We perform a multichannel fit () when possible by including single energy data from process, and we fit both multipoles for the same angular momentum at the same time in the coupled-multipole fit (). The regular background part is represented by three Pietarinen expansion series, all free parameters are fitted. The first Pietarinen expansion with branch-point is restricted to an unphysical energy range and represents all left-hand cut contributions. The next two Pietarinen expansions describe the background in the physical range with branch-points and respecting the analytic properties of the analyzed partial wave. The second branch-point is mostly fixed to the elastic channel branch-point, the third one is either fixed to the dominant channel threshold, or left free. Thus, only rather general physical assumptions about the analytic properties are made like the number of poles and the number and the position of branch-points, and the simplest analytic function with a set of poles and branch-points is constructed.
In the compilation of our results we show the results of four fits: a) the BnGa coupled channel fit to the complete data base including the energy independent solutions for and presented here; b) a single-channel L+P fit to the energy independent solution for () ; c) a single-channel L+P fit to the energy independent solution for (); and d) a multi-channel L+P fit to the energy independent solution for and ().
4.2 L+P Fits
We have fitted the partial wave from the energy independent amplitude for the reaction
in a fit. A was obtained for the 28 data points with 23 parameters. We needed two poles, one at 1667 MeV and second one at 1910 MeV. Due to the low-statistics of the data, the results from the single-channel fit show large errors.
The 48 data points on the multipole from required only one pole close to 1900 MeV. The strong peak at low mass of the imaginary part of the multipole is reproduced by the function with a branching point at the threshold. Note that the lowest mass bin for the multipole starts at 1700 MeV, significantly above the mass. The data were described with a and 19 parameters in a fit. Compared to the pion-induced reaction, the errors on the higher-mass resonance (at 1900 MeV) are considerably reduced.
The real part of the pole positions of the resonance are nicely consistent when the three values are compared, the imaginary part is likely too narrow in the L+P fit. The magnitudes of the inelastic pole residue are consistent at the level when the BnGa and CC L+P fits are compared. The phases, however, seem to be inconsistent.
The pole positions are well defined with acceptable errors and consistent when the four analyses are compared, only the single-channel L+P fit to photoproduction data returns a slightly too narrow width. All four analyses yield compatible magnitudes of the inelastic pole residues, the phases disagree at the level. The magnitudes and the phases of the multipole determined by the BnGa fit agree well with the values of the L+P fits within the rather large uncertainties. Note that the errors in the CC L+P and BnGa fits have different origins: The L+P errors are of a statistical nature, the BnGa errors are derived from the spread of results of a variety of different fits. Both approaches establish the need for and unquestionably require .
We have fitted the -wave using the P energy independent amplitude for the reaction and the multipole from . The first data set required two poles. The first pole was located near 1700 MeV, the second one was found near 2100 MeV even though with large error bars: the admitted range covers masses from to MeV. The photoproduction data required only one pole close to 1900 MeV. The CC L+P fit to both data sets was performed with two poles.